How to boolean with non manifold geometry?
I will demonstrate my approach on Suzanne so that it is a bit more complex than say, a cube.


Setup the "knife cutter". For this I use a simple mesh with 3 vertices. Notice that the cursor is set to the pivot point for easy rotation control.





Perform the knife project.





Clean up the cut out vertices. Notes: use wireframe mode so the selection goes "through" the mesh, don't select the verts along the cut line...just the ones inside.